Rare tau decays via exchange of on-shell almost degenerate Majorana neutrinos, τ^∓ → π^∓N_j → π^∓μ^∓π^± and τ^∓ → π^∓N_j → π^∓μ^±π^∓

We investigate rare decays of tau leptons that occur via exchange of heavy on-shell neutrinos N_j (j = 1, 2). These neutrinos can be either Dirac or Majorana, and are considered to be almost degenerate in mass. The decays can thus be either lepton number conserving (LNC), τ^∓ → π^∓N_j → π^∓μ^∓π^±, or lepton number violating (LNV), τ^∓ → π^∓N_j → π^∓μ^±π^∓. If neutrinos are Dirac, only LNC decays are possible. If they are Majorana, both LNC and LNV are possible. We derive the corresponding expressions for the effective decay widths ${\mathrm{\Gamma }}_{\mathrm{eff},\mp }^{\left(\mathrm{X}\right)}$ \( {\Gamma}_{\textrm{eff},\mp}^{\left(\textrm{X}\right)} \) (X=LNC, LNV) of these rare decays, where we account for N₁-N₂ overlap and oscillation effects and for the finite detector length effects. We then numerically evaluate these decay widths as well as the related CP violation asymmetry width $\text{∆}{\mathrm{\Gamma }}_{\mathrm{CP}}^{\left(\mathrm{X}\right)}=\left({\mathrm{\Gamma }}_{\mathrm{eff},-}^{\left(\mathrm{X}\right)}-{\mathrm{\Gamma }}_{\mathrm{eff},+}^{\left(\mathrm{X}\right)}\right)$ \( \Delta {\Gamma}_{\textrm{CP}}^{\left(\textrm{X}\right)}=\left({\Gamma}_{\textrm{eff},-}^{\left(\textrm{X}\right)}-{\Gamma}_{\textrm{eff},+}^{\left(\textrm{X}\right)}\right) \) . We conclude that for certain, presently allowed, ranges of the heavy-light neutrino mixing parameters, such decays and asymmetries could be observed in Belle II experiment.